Welcome to the forum!
If the water doesn't drain out of the pot, the plant's roots will rot. No way to avoid that.
The "green" you see on the concrete is probably algae. It should be easy to kill it with a little algacide, such as is made for fountains, birdbaths, ponds and aquariums. It works really well in my ponds and birdbaths.
I would just put a capful of the algicide in a bucket of water and pour it over the green area. You may need to use more of the algacide and water solution, depending on the size of the area you need to treat. Another method would be to put the solution in a plant sprayer and spray it on the green area. It might be necessary to scrub the dead algae off the concrete after the first treatment, but regular follow-up treatments, maybe once a month or so, should prevent more algae from growing. It's what I would do, anyway.